Casimir effect with Gaussian regulator

[Silviu]

Homework Statement

Calculate the Casimir force in 1D using a Gaussian regulator.

Homework Equations

The Attempt at a Solution

I reached a point where I need to evaluate a sum of the form $$\sum_n n e^{-\epsilon^2n^2}$$ Can someone help me? I didn't really find anything useful online. I thought of turning this into a gaussian integral, and I would get something of the form $\frac{1}{\epsilon}\frac{\sqrt{\pi}}{2}$, but I don't get the answer I need using this.

 

[MathematicalPhysicist]

Check it in Voja's problem book under regularization.

 

[Silviu]

I didn't find it there...

 

[MathematicalPhysicist]

Try using Abel's summation formula:
https://en.wikipedia.org/wiki/Abel's_summation_formula